problem
stringlengths 18
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|---|---|---|
Find the area of the region bounded by the graphs of \((x+2ay)^2 = 16a^2\) and \((2ax-y)^2 = 4a^2\), where \(a\) is a positive real number.
|
\frac{32a^2}{1 + 4a^2}
| 0.083333 |
Sally is 20% younger than Danielle, Adam is 30% older than Mary, and Mary is 25% younger than Sally. Given that the sum of their ages is 60 years, determine Sally's next birthday age.
|
16
| 0.5 |
Let $x = .123456789101112....109810991100$, where the digits are obtained by writing the integers $1$ through $1099$ in order. Determine the $2500$th digit to the right of the decimal point.
|
8
| 0.666667 |
Given the rug display featuring sections in three distinct colors, where the areas of these sections form an arithmetic progression, and the width of the center section is 2 feet, and each of the two colored rings around the center section extends outward by 2 feet on all sides, determine the length in feet of the center section.
|
4
| 0.416667 |
A woman buys an apartment for $20,000 and rents it out. She sets aside 10% of each month's rent for maintenance and pays $460 a year in taxes. Determine the monthly rent she needs to charge to realize a 6% return on her investment annually.
|
153.70
| 0.583333 |
Calculate the number of positive integers less than 1200 that are divisible by neither 6 nor 8.
|
900
| 0.833333 |
Moe, Loki, and Nick are companions with distinct amounts of money. Moe gives Ott one-fifth of his money, Loki gives Ott one-third of his money, and Nick gives Ott one-half of his money. If each gave Ott the same amount of money, find the fraction of the group's total money that Ott now possesses.
|
\frac{3}{10}
| 0.916667 |
A 10x10 arrangement of alternating black and white squares has a black square $R$ in the second-bottom row and a white square $S$ in the top-most row. Given that a marker is initially placed at $R$ and can move to an immediately adjoining white square on the row above either to the left or right, and the path must consist of exactly 8 steps, calculate the number of valid paths from $R$ to $S$.
|
70
| 0.75 |
In how many ways can 10003 be written as the sum of two primes?
|
0
| 0.083333 |
Find the value of $(3(3(3(3(3(3+2)+2)+2)+2)+2)+2)$.
|
1457
| 0.416667 |
Given a circle of radius \(3\), find the area of the region consisting of all line segments of length \(6\) that are tangent to the circle at their midpoints.
|
9\pi
| 0.083333 |
Given Jo and Blair take turns counting numbers in a sequence where each number is two more than the last number said by the other person, and Jo starts by saying $2$, determine the $30^{\text{th}}$ number said.
|
60
| 0.833333 |
Given a warehouse stores 110 containers of oranges, where each container has at least 98 oranges and at most 119 oranges, determine the largest integer n such that there must be at least n containers having the same number of oranges.
|
5
| 0.583333 |
Let \( f \) be a function defined by \( f\left(\frac{x}{3}\right) = x^2 + x + 1 \). Calculate the sum of all values of \( z \) for which \( f(3z) = 4 \).
|
-\frac{1}{9}
| 0.666667 |
Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $3$, $2$, and $4$, respectively, and constant terms $2$, $3$, and $6$, respectively. If $P(z)$ and $Q(z)$ each have $z=-1$ as a root, determine the minimum possible value of the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) = R(z)$.
|
1
| 0.5 |
The set of all real numbers \(y\) for which the expression \(\log_{10}(\log_{9}(\log_{8}(\log_{7}(y^{2}))))\) is defined. Represent this set as \(\{y \mid y > d\}\). Determine the value of \(d\).
|
2401
| 0.75 |
Given 300 swimmers compete in a 50-meter freestyle event with 8 lanes, each allowing 8 swimmers to compete simultaneously, and the top two swimmers in each race advance to the next round, determine the number of races required to find the champion swimmer.
|
53
| 0.75 |
The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class.
|
71.47
| 0.666667 |
If one side of a triangle is $20$ inches and the opposite angle is $45^{\circ}$, calculate the diameter of the circumscribed circle.
|
20 \sqrt{2}\text{ inches}
| 0.916667 |
There are 30 tiles in the box, containing 100 edges in total. Find the number of square tiles in the box.
|
10
| 0.5 |
The average (arithmetic mean) age of a group consisting of doctors and lawyers is 47. If the doctors average 45 years old and the lawyers 55 years old, calculate the ratio of the number of doctors to the number of lawyers.
|
4
| 0.833333 |
Given the sets of fractions $S_1$ and $S_2$ defined as $S_1$ includes fractions from $\dfrac{1}{10}$ up to $\dfrac{9}{10}$ incrementing by $1$ in the numerators, and $S_2$ is created by the constant $\dfrac{20}{10}$ repeated four times, calculate the sum of all fractions from $S_1$ and $S_2$.
|
12.5
| 0.666667 |
Evaluate the value of the expression $\frac{121^2 - 13^2}{91^2 - 17^2} \cdot \frac{(91-17)(91+17)}{(121-13)(121+13)}$.
|
1
| 0.75 |
Given that one root of $ax^2 + bx + c = 0$ is triple the other, determine the relationship between the coefficients $a, b, c$.
|
3b^2 = 16ac
| 0.916667 |
Calculate the sum of the sequence $3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + 5\left(1-\dfrac{1}{5}\right) + \cdots + 11\left(1-\dfrac{1}{11}\right)$.
|
54
| 0.333333 |
Given a square with a side length of 10 cut into two unequal rectangles, one of which is twice the area of the other, find the dimensions of the smaller rectangle.
|
\left(\frac{10}{3}, 10\right)
| 0.333333 |
A positive integer divisor of $10!$ is chosen at random. Calculate the probability that the divisor chosen is a perfect square, expressed as a simplified fraction $\frac{m}{n}$, and find the sum of the numerator and denominator.
|
10
| 0.916667 |
Find the smallest integer value of $N$ so that when $(a+b+c+d+e+1)^N$ is expanded and like terms are combined, the expression contains exactly $3003$ terms that include all five variables $a, b, c, d,$ and $e$, each to some positive power.
|
15
| 0.416667 |
Consider the sequence $2, -4, 6, -8, 10, -12, \ldots,$ where each term $a_n$ in the sequence is calculated by $(-1)^n \cdot 2n$. Find the average of the first $300$ terms of this sequence.
|
-1
| 0.833333 |
Consider the set $\{45, 52, 87, 90, 112, 143, 154\}$. How many subsets containing three different numbers can be selected from this set so that the sum of the three numbers is odd?
|
19
| 0.333333 |
Suppose the estimated $30$ billion dollar cost to send a person to Mars and another $10$ billion dollars to establish a base on the Moon is shared equally by $200$ million people in a country. Calculate each person's share.
|
200
| 0.916667 |
Define a two-digit positive integer as snuggly if it is equal to the sum of its nonzero tens digit, the cube of its units digit, and 5. How many two-digit positive integers are snuggly?
|
0
| 0.833333 |
Determine the least possible value of \((x+2)(x+3)(x+4)(x+5) + 2024\) where \(x\) is a real number.
|
2023
| 0.75 |
Let's say Cagney can frost a cupcake every 25 seconds, and Lacey can frost a cupcake every 35 seconds. Calculate the number of cupcakes they can frost in 10 minutes if they work together.
|
41
| 0.916667 |
Calculate the square of $7 - \sqrt[3]{x^3 - 49}$.
|
49 - 14\sqrt[3]{x^3 - 49} + (\sqrt[3]{x^3 - 49})^2
| 0.666667 |
Determine the total cost David had to pay for his cell phone plan, given a basic monthly fee of $30, a cost of $0.10 per text message, and a cost of $0.15 for each minute over a standard usage of 20 hours, after sending 150 text messages and using the phone for 21 hours.
|
54
| 0.916667 |
What is the sum of the mean, median, and mode of the numbers $1,2,1,4,3,1,2,4,1,5$?
|
5.4
| 0.25 |
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 60?
|
4087
| 0.833333 |
Successive discounts of $15\%$ and $25\%$ are equivalent to a single discount.
|
36.25\%
| 0.916667 |
Given that $a \oplus b$ means $4a + 2b$, find the value of $y$ if $3 \oplus (4 \oplus y) = -14$.
|
-14.5
| 0.083333 |
Given that $\angle \text{CBD}$ is a right angle, the sum of the angles around point B is $180^\circ$ and $\angle \text{ABD} = 30^\circ$, find the measure of $\angle \text{ABC}$.
|
60^\circ
| 0.666667 |
Leah places six ounces of tea into a twelve-ounce cup and six ounces of milk into a second cup of the same size. She then pours one-third of the tea from the first cup to the second and, after stirring thoroughly, pours one-fourth of the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now milk?
|
\frac{1}{4}
| 0.416667 |
Given the expression $n = \frac{2kcd}{c+d}$, solve for $d$.
|
\frac{nc}{2kc - n}
| 0.833333 |
Given $a$ and $b$ are even single-digit positive integers chosen independently and at random, find the probability that the point $(a,b)$ lies above the parabola $y = ax^2 + bx$.
|
0
| 0.75 |
Given that the sum of two or more consecutive positive integers is 20, find the number of sets.
|
1
| 0.333333 |
Find the value of $y$ that satisfies the equation $16^{-3} = \frac{4^{72/y}}{4^{42/y} \cdot 16^{25/y}}$.
|
\frac{10}{3}
| 0.75 |
For how many integers $x$ is the point $(x, -x)$ inside or on the circle of radius $12$ centered at $(3, 3)$?
|
15
| 0.833333 |
Calculate the product $6 \times 0.5 \times \frac{3}{4} \times 0.2$.
|
\frac{9}{20}
| 0.833333 |
The parabolas \( y = ax^2 - 3 \) and \( y = 5 - bx^2 \) intersect the coordinate axes in exactly four points, and these four points form the vertices of a kite with an area of 18. Determine \( a + b \).
|
\frac{128}{81}
| 0.75 |
Given a bag of popcorn kernels containing $\frac{3}{4}$ white kernels and $\frac{1}{4}$ yellow kernels, and with $\frac{2}{5}$ of the white kernels and $\frac{3}{4}$ of the yellow kernels popping when placed in a popper, determine the probability that a kernel was white if it was selected at random and successfully popped.
|
\frac{8}{13}
| 0.916667 |
Given the expression \(125\sqrt[3]{16}\sqrt[6]{64}\), evaluate the logarithm to the base \(5\).
|
3 + \frac{7}{3}\log_5(2)
| 0.333333 |
In a bucket, there are $34$ red balls, $25$ green balls, $23$ yellow balls, $18$ blue balls, $14$ white balls, and $10$ black balls. Find the minimum number of balls that must be drawn from the bucket without replacement to guarantee that at least $20$ balls of a single color are drawn.
|
100
| 0.333333 |
In trapezoid EFGH, the bases EF and GH are not equal, where EF = 10 units and GH = 14 units. The lengths of the non-parallel sides are EG = 7 units and FH = 7 units. Calculate the perimeter of trapezoid EFGH.
|
38
| 0.833333 |
Two concentric circles have radii in the ratio $2:5$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 8$, calculate the radius of the larger circle.
|
10
| 0.5 |
An equilateral triangle is entirely painted blue. Each iteration, the middle ninth of each blue region turns green. After three iterations, what fractional part of the original area of the blue triangle remains blue.
|
\frac{512}{729}
| 0.75 |
A set $T$ of points in the $xy$-plane is symmetric about both coordinate axes and the line $y=-x$. If $(1,4)$ is in $T$, calculate the smallest number of points in $T$.
|
8
| 0.75 |
A circle is inscribed in a triangle with side lengths $9, 12$, and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $u$ and $v$, with $u<v$. Find the ratio $u:v$.
|
\frac{1}{2}
| 0.5 |
Given that $\boxed{n}$ is defined to be the sum of the positive factors of $n$, find $\boxed{\boxed{13}}$.
|
24
| 0.916667 |
Find the largest power of 2 that divides $17^4 - 13^4$.
|
16
| 0.916667 |
Given Gabriel's weight of 96 pounds, along with the weights of his quadruplet brothers, 10, 10, 12, and 18 pounds, calculate the difference between the average and median weights of the five children.
|
17.2
| 0.666667 |
Given the list of the first 12 positive integers such that for each \(2 \le i \le 12\), either \(b_i+1\) or \(b_i-1\) or both appear somewhere before \(b_i\) in the list, find the number of such lists.
|
2048
| 0.083333 |
A square with area $16$ is inscribed in a square with area $25$, where each vertex of the smaller square touches a side of the larger square. One vertex of the smaller square divides a side of the larger square into two segments, and one length is three times the other length. Find the product of the lengths of the two segments.
|
\frac{75}{16}
| 0.666667 |
Given that Star lists the whole numbers from $1$ to $50$ once, and Emilio copies Star's numbers replacing each occurrence of the digit '3' by the digit '2', calculate the difference between Star's sum and Emilio's sum of the copied numbers.
|
105
| 0.25 |
A rectangle with dimensions $10$ by $4$ shares the same center with an ellipse having semi-major axis lengths of $3$ and semi-minor axis lengths of $2$. Calculate the area of the region common to both the rectangle and the ellipse.
|
6\pi
| 0.416667 |
Given 1 brown tile, 2 purple tiles, 2 red tiles, and 3 yellow tiles, determine the number of distinguishable arrangements of these tiles in a row from left to right.
|
1680
| 0.833333 |
A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Calculate the value of $d$ to the nearest tenth.
|
0.3
| 0.666667 |
A school has $100$ students divided among $5$ classes taught by $5$ teachers, with each teacher teaching one class. The student distribution across the classes is $40, 40, 10, 5, 5$. Compute the value of $t-s$, where $t$ is the expected number of students per class when choosing a teacher at random and $s$ is the expected number of students per class when choosing a student at random.
|
-13.5
| 0.416667 |
Given a palindrome between $1000$ and $10,000$ is chosen at random, find the probability that it is divisible by both 3 and 11.
|
\frac{1}{3}
| 0.833333 |
Given that $\frac{4}{7}$ of the marbles are blue, if the number of red marbles is tripled while the number of blue marbles remains the same, calculate the resulting fraction of marbles that will be red.
|
\frac{9}{13}
| 0.833333 |
Evaluate the expression $(4+5)(4^2+5^2)(4^4+5^4)(4^8+5^8)(4^{16}+5^{16})(4^{32}+5^{32})(4^{64}+5^{64})$.
|
5^{128} - 4^{128}
| 0.583333 |
In a room where $3/7$ of the people are wearing gloves, and $5/6$ of the people are wearing hats, determine the minimum number of people in the room wearing both a hat and a glove.
|
11
| 0.916667 |
A lemming starts at a corner of a rectangular area measuring 8 meters by 15 meters. It dashes diagonally across the rectangle towards the opposite corner for 11.3 meters. Then the lemming makes a $90^{\circ}$ right turn and sprints upwards for 3 meters. Calculate the average of the shortest distances to each side of the rectangle.
|
5.75
| 0.333333 |
A ticket to a school concert costs $x$ dollars, where $x$ is a whole number. A group of 8th graders buys tickets costing a total of $36$, and a group of 9th graders buys tickets costing a total of $63$. Determine the number of values for $x$ that are possible.
|
3
| 0.916667 |
Given the function $g$ defined by the table
\[
\begin{tabular}{|c||c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline g(x) & 5 & 3 & 1 & 2 & 4 \\ \hline \end{tabular}
\]
calculate the value of $u_{2023}$ given that $u_0=5$ and $u_{n+1} = g(u_n)$ for $n \ge 0$.
|
3
| 0.916667 |
How many even positive 4-digit integers are divisible by 5 but do not contain the digit 5?
|
648
| 0.666667 |
Given the product $\frac{4}{3}\times\frac{5}{4}\times\frac{6}{5}\times\cdots\times\frac{2010}{2009}$, find the value of this product.
|
670
| 0.5 |
Given a triangle with vertices at $A = (2, 2)$, $B = (6, 3)$, and $C = (5, 6)$ plotted on a $7 \times 6$ grid, calculate the fraction of the grid that is covered by the triangle.
|
\frac{13}{84}
| 0.916667 |
Given Alice has 30 apples, find the number of ways she can share them with Becky and Chris so that each of the three people has at least three apples.
|
253
| 0.583333 |
Calculate the value of $3000 \cdot (3000^{2500}) \cdot 2$.
|
2 \cdot 3000^{2501}
| 0.5 |
Given a cylindrical water tank that stands 50 meters high and can hold 200,000 liters of water, and a miniature water tank that holds only 0.2 liters, determine the height of the miniature water tank.
|
0.5
| 0.333333 |
If the sum of the first $20$ terms and the sum of the first $50$ terms of a given arithmetic progression are $150$ and $20$, respectively, then find the sum of the first $70$ terms.
|
-\frac{910}{3}
| 0.25 |
A belt is fitted without slack around two circular pulleys with radii of $10$ inches and $6$ inches. The belt runs externally and does not cross over itself. The distance between the points where the belt contacts with the pulleys is $30$ inches. Find the distance between the centers of the two pulleys.
|
2\sqrt{229}
| 0.083333 |
Given quadrilateral $ABCD$ is divided into four triangles by point $P$, solve for $\alpha$ in terms of $\beta$, $\gamma$, and $\delta$.
|
360^\circ - \beta - \gamma - \delta
| 0.916667 |
Jasmine has a large bottle that can hold 450 milliliters and a small bottle that holds 45 milliliters. She obtained an extra large bottle with a capacity of 900 milliliters. Determine the minimum total number of bottles required to completely fill the extra large bottle.
|
2
| 0.916667 |
Given positive integers $a$ and $b$ that are each less than 10, calculate the smallest possible value for $3 \cdot a - a \cdot b$.
|
-54
| 0.166667 |
The taxi fare in Gotham City is $3.00 for the first $\frac12$ mile and additional mileage charged at the rate $0.25 for each additional 0.1 mile, and a $3 tip is planned. Calculate the number of miles that can be ridden for $15.
|
4.1
| 0.75 |
Points $A, B, C,$ and $D$ lie on a line in that order, with $AB = CD$ and $BC = 8$. Point $E$ is not on the line, and $BE = CE = 8$. Find $AB$ given that the perimeter of $\triangle AED$ is three times the perimeter of $\triangle BEC$.
|
\frac{40}{3}
| 0.083333 |
In a triangle with sides 50, 110, and 120 units, an altitude is dropped upon the side of length 120. Determine the length of the larger segment cut off on this side by the altitude.
|
100
| 0.333333 |
Sharon's car uses a gallon of gas every 40 miles, and her gas tank holds 16 gallons when it is full. One day, Sharon started with a full tank of gas, drove 480 miles, bought 6 gallons of gas, and continued driving to her destination. When she arrived, her gas tank was a quarter full. Calculate the total number of miles Sharon drove that day.
|
720
| 0.916667 |
The student locker numbers at Liberty High are numbered consecutively starting from locker number $1$. Each digit costs three cents. If it costs $206.91 to label all the lockers, how many lockers are there?
|
2001
| 0.25 |
The sum of seven test scores has a mean of 84, a median of 85, and a mode of 88. Calculate the sum of the three highest test scores.
|
264
| 0.333333 |
Determine how many non-prime numbers exist where the digits sum to 7, and each digit in the number is greater than 1.
|
5
| 0.166667 |
A cylindrical water tank has a radius $R$ of $10$ inches and a height $H$ of $5$ inches. Determine the value of $x$ such that the increase in volume, when the radius is increased by $x$ inches, is twice the increase in volume when the height is increased by $x$ inches.
A) $x = 0$
B) $x = 10$
C) $x = 20$
D) $x = 40$
|
C) \ x = 20
| 0.75 |
Let $Q$ equal the product of 1,234,567,890,123,456,789 and 987,654,321,098,765 multiplied by 123. Calculate the number of digits in $Q$.
|
36
| 0.25 |
Quadrilateral $EFGH$ is a rhombus with a perimeter of $80$ meters. The length of diagonal $\overline{EG}$ is $36$ meters. Calculate the area of rhombus $EFGH$.
|
72\sqrt{19}
| 0.916667 |
Given a regular decagon, a triangle is formed by connecting three randomly chosen vertices of the decagon. Calculate the probability that exactly one of the sides of the triangle is also a side of the decagon.
|
\frac{1}{2}
| 0.5 |
A paper triangle with sides of lengths $4, 4, 6$ inches is folded so that point $A$ falls on point $C$ (where $AC = 6$ inches). What is the length, in inches, of the crease?
|
\sqrt{7}
| 0.166667 |
Three congruent circles are centered at points $A$, $B$, and $C$ such that each circle passes through the center of the other two circles. The line containing both $A$ and $B$ is extended to intersect the circles at points $D$ and $E$ respectively. The circles centered at $A$ and $C$ intersect at a point $F$ distinct from where they intersect with the circle centered at $B$. Calculate the degree measure of $\angle DFE$.
|
120^{\circ}
| 0.833333 |
Given a 7-period day, and the condition that no two mathematics courses can be taken in consecutive periods, and the first and last periods cannot be math courses, calculate the number of ways a student can schedule 3 mathematics courses -- algebra, trigonometry, and calculus.
|
6
| 0.083333 |
Given a town's population increased by $1,500$ people, and then this new population decreased by $15\%$, and after these changes the town had $45$ fewer people than it did before the increase, determine the original population of the town.
|
8800
| 0.916667 |
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